Question

It is an unfair coin with unknown probability of head and tail. Can you generate even odds using this coin?

Answer 1

Answer:

Yes, it can be generated if $p = \frac{n}{n+1}$, where n is a positive even integer.

Step-by-step explanation:

An odds for an event, is the probability or chance that this event will occur, divided by the probability that it will not occur.

Assume that the probability of getting tail when throwing the coin is $p$, therefore, the probability of getting a head is $(1-p)$ and the odds for the event of getting tail is $\frac{p}{1-p}$.

If $k$ is a positive integer, and we assume that the odds of getting a tail are even, then:

$\frac{p}{1-p} = 2k$

$p = 2k - 2kp$

$p = \frac{2k}{2k+1}$

Conclusion: if $p = \frac{2k}{2k+1}$, for some positive integer $k$, even odds are generated. For example: $p = 2/3, 4/5, 6/7, ..$.